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Theorem cntzidss 14829
Description: If the elements of  S commute, the elements of a subset  T also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypothesis
Ref Expression
cntzmhm.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
cntzidss  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  T  C_  ( Z `  T
) )

Proof of Theorem cntzidss
StepHypRef Expression
1 simpr 447 . 2  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  T  C_  S )
2 simpl 443 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  S  C_  ( Z `  S
) )
3 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
4 cntzmhm.z . . . . . 6  |-  Z  =  (Cntz `  G )
53, 4cntzssv 14820 . . . . 5  |-  ( Z `
 S )  C_  ( Base `  G )
62, 5syl6ss 3204 . . . 4  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  S  C_  ( Base `  G
) )
73, 4cntz2ss 14824 . . . 4  |-  ( ( S  C_  ( Base `  G )  /\  T  C_  S )  ->  ( Z `  S )  C_  ( Z `  T
) )
86, 7sylancom 648 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  ( Z `  S )  C_  ( Z `  T
) )
92, 8sstrd 3202 . 2  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  S  C_  ( Z `  T
) )
101, 9sstrd 3202 1  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  T  C_  ( Z `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165   ` cfv 5271   Basecbs 13164  Cntzccntz 14807
This theorem is referenced by:  gsumzres  15210  gsumzf1o  15212  gsumzaddlem  15219  gsumzadd  15220  gsumzsplit  15222  gsumconst  15225  gsumpt  15238  dprdfadd  15271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-cntz 14809
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